Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 A Single Population Mean Using the Normal Distribution
3 A Single Population Mean Using the Student t Distribution
4 A Population Proportion
5 Conclusion
This is, in essence, getting us closer to inferential statistic, where we can actually infer meaning from the characteristics of our sample
The part that we need to add are called confidence interval
This becomes practically interpreted as 95% of the sample statistics taken from this population with the same sample size (\(n\)) would give us a statistic within this range 95% of the time
For example: I have a point estimate mean statistic (\(\bar{x}\)) of 100, a sample size (\(n\)) of 40, and a population parameter standard deviation (\(\sigma\)) of 10
Agenda
1 Overview and Introduction
2 A Single Population Mean Using the Normal Distribution
3 A Single Population Mean Using the Student t Distribution
4 A Population Proportion
5 Conclusion
Agenda
1 Overview and Introduction
2 A Single Population Mean Using the Normal Distribution
3 A Single Population Mean Using the Student t Distribution
4 A Population Proportion
5 Conclusion
In the case that we don't know the population standard deviation, we can use the t-distribution in the following way
\(EBM = (t_\frac{\alpha}{2})(\alpha{s}{\sqrt{n}})\)
Agenda
1 Overview and Introduction
2 A Single Population Mean Using the Normal Distribution
3 A Single Population Mean Using the Student t Distribution
4 A Population Proportion
5 Conclusion
Often, we are interested not just in the confidence interval of means, but also of percentages and proportions
First, we need to identify a situations that is more proportion-based than mean-based
Because of the underlying binomial, we can take a proportion as: \(P' = \frac{X}{N}\)
In the scenario that \(n\) is particularly large and \(p\) is not close to zero or one, we can actually treat this as normal like: \(X \sim N(np, \sqrt{npq})\)
This all eventually works down to \(\frac{X}{n} = P' \sim N(\frac{np}{n}, \frac{\sqrt{npq}}{n})\)
One can calculate error bound for a proportion (EBP) to work towards a sort of confidence interval, but for proportions, just like we did for means
Agenda
1 Overview and Introduction
2 A Single Population Mean Using the Normal Distribution
3 A Single Population Mean Using the Student t Distribution
4 A Population Proportion
5 Conclusion
I am a big advocate of the value of confidence intervals in a lot of situations - I think they appropriately capture that these statistics really are just estimates/guesses!
In my opinion, it is pretty much always appropriate to report confidence intervals alongside the point estimates of just about anything you calculate - it increases information and transparency
Confidence intervals add valuable information about the relative spread and inaccuracy of sample statistics in means and in proportions
Confidence intervals pretty much always become more narrow with larger sample size, continuing the trends demonstrated with the central limit theorem and the law of large numbers
We introduced the t-distribution as a valuable way to calculate confidence intervals for means, even when lacking the population standard deviation
Module 8 Lecture - Multiple Comparisons Under Factorial ANOVA || Analysis of Variance